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Friday, November 4, 2016

Mathematical Milk and the U.S. Presidential Election

Keith Devlin mails his completed election ballot. What does math have to say about his act?

With the United States is the final throes of a presidential election, my mind naturally turned to the
decidedly tricky matter of election math. Voting provides a great illustration of how
mathematics – which rules supreme, yielding accurate and reliable answers to precise
questions, in the natural sciences and engineering – can lead us astray when we try to
apply it to human and social activities.

A classic example is how we count votes in an election, the topic of an earlier Devlin’sAnglepost, in November, 2000. In that essay, I looked at how different ways to tally votes
could affect the imminent Bush v. Gore election, at the time blissfully unaware of how
chaotic would be the process of counting votes and declaring a winner on that particular
occasion. The message there was, particularly in the kinds of tight race we typically see
today, the different ways that votes can be tallied can lead to very different results.

Everything I said back then remains just as valid and pertinent today (mathematics is
like that), so this time I’m going to look at another perplexing aspect of election math: why do we make the effort to vote? After all, while elections are sometimes decided by
a small number of votes, it is unlikely in the extreme that an election on the scale of a
presidential election will hang on the decision of a single voter. Even if it did, that
would be well within the range of procedural error, so it makes no difference if any one
individual votes or not.

To be sure, if a large number of people decide to opt out, that can affect the outcome. But there is no logical argument that takes you from that observation to it being important for a single individual to vote. This state of affairs is known as the Paradox of Voting, or sometimes Downs Paradox. It is so named after Anthony Downs, a political economist whose 1957 book An Economic Theory of Democracy examined the conditions under which (mathematical) economic theory could
be applied to political decision-making.

On the face of it, Downs’ analysis does lead to a paradox. Economic theory tells us that rational beings make decisions based on expected benefit (a notion that can be
made numerically precise). That approach works well for analyzing, say, why people buy insurance year after year, even though they may never submit a claim. The theory
tells you that the expected benefit is greater than the cost; so it is rational to buy
insurance. But when you adopt the same approach to an election, you find that,
because the chance of exercising the pivotal vote in an election is minute compared to
any realistic estimate of the private individual benefits of the different possible
outcomes, the expected benefits of voting are less than the cost. So you should opt out.
[The same observation had in fact been made much earlier, in 1793, by Nicolas de
Condorcet, but without the theoretical
backing that Downs brought to the issue.]

Yet, many otherwise sane, rational citizens do not opt out. Indeed, society as a whole
tends to look down on those who do not vote, saying they are not "doing their part." (In
fact, many countries make participation in a national election obligatory, but that is a
separate, albeit related, issue.)

So why do we (or at least many of us) bother to vote? I can make the question even
more stark, and personal. Suppose you have intended to "do your part" and vote. You
wake up on election morning with a sore throat, and notice that it is raining heavily.
Being numerically able (as all Devlin’s Angle readers must be), you say to yourself, "It
cannot possibly affect the result if I just stay at home and nurse my throat. I was
intending to vote, after all. Changing my mind about voting at the last minute
cannot possibly influence anyone else. Especially if I don’t tell anyone." The math and
the logic, surely, are rock solid. Yet, professional mathematician as I am, I would
struggle out and cast my vote. And I am sure many Devlin’s Angle readers would too –
most of them, I would suspect.

So what is going on? We can do the math. We are good logical thinkers. Why don’t we
act according to that reasoning? Are we conceding that mathematics actually isn’t that
useful? [SPOILER: Math is useful; but only when applied with a specific purpose in
mind, and chosen/designed in a way that makes it appropriate for that purpose.]

Which brings me to my main point. To make it, let me switch for a moment from
elections to the Golden Ratio. In April 2015, the magazine Fast Company Design
published an article titled "The Golden Ratio: Design’s Biggest Myth," in which I was
quoted at length. (The author also drew heavily on a Devlin’s Angle post of mine from
May 2007.)

With a readership much wider than Devlin’s Angle, over the years the Fast Company Design piece has generated a fair amount of correspondence from people beyond mathematics
academia, often designers who have not been able to overcome drinking Golden Ratio
Kool-Aid during their design education. One recent email came, not from a designer but
a high school math teacher, who objected to a statement the article quoted me
(accurately) as saying, “Strictly speaking, it's impossible for anything in the real-world to
fall into the golden ratio, because it’s an irrational number.” The teacher had, it was at
once clear to me, drunk not just Golden Ratio Kool-Aid, but Math Kool-Aid as well.

In the interest of full disclosure, let me admit that, in the early part of my career as a
mathematics expositor, I was as guilty as anyone of distributing both Golden Ratio Kool-Aid and Math Kool-Aid, to whoever would drink it. But, as a committed scientist, when
presented with evidence to the contrary, I re-examined my thinking, admitted I had been
wrong, and started to push better, more honest products, which I will call Golden Ratio
Milk and Mathematical Milk. I described Golden Ratio Milk in my 2007 MAA post and
peddled it more in that Fast Company Design interview. Here I want to talk about Mathematical
Milk.

The reason why the Golden Ratio’s irrationality prevents its use in, say architecture, is
that the issue at hand involves measurement. Measurement requires fixing a unit of
measure – a scale. It doesn’t matter whether it is meters or feet or whatever, but once
you have fixed it, that is what you use. When you measure things, you do so to an
agreed degree of accuracy. Perhaps one or two decimal places. Almost never to more
than maybe twenty decimal places, and that only in a few instances in subatomic
physics. So it terms of actual, physical measurement, or manufacturing, or building, you
never encounter objects to which a numerical measurement has more than a few
decimal places. You simply do not need a number system that has fractions with
denominator much greater than, say, 1,000,000, and generally much less than that.

Even if you go beyond physical measurement, to the theoretical realm where you
imagine having an unlimited number decimal places available, you will still be in the
domain of the rational numbers. Which means the Golden Ratio does not arise.
Irrational numbers arise to meet mathematical needs, not the requirements of
measurement. They live not in the physical world but in the human imagination. (Hence
my Fast Company Design quote.) It is important to keep that distinction clear in our minds.

The point is, when we abstract from our experiences of the world around us, to create
mathematical models, two important things happen. A huge amount of information is
lost; and there is a significant gain in precision. The two are not independent. If we want
to increase the precision, we lose more information, which means that our model has
less in common with the real world it is intended to represent. Moreover, when we
construct a mathematical model, we do so with a particular question, or set of questions
in mind.

In astronomy and physics, and related domains such as engineering, all of this turns out
to be not too problematic. For example, the simplistic model of the Solar System as a
collection of point-masses orbiting around another, much heavier, point-mass, is
extremely useful. We can formulate and solve equations in that model, and they turn out
to be very useful. At least they turn out to be useful in terms of the goal questions,
initially in this case predicting where the planets will be at different times of the year.
The model is not very helpful in telling us what the color of each planet’s surface is, or
even if it has a surface, both of which are certainly precise, scientific questions.

When we adopt a similar approach to model money supply or other economic
phenomena, we can obtain results that are, mathematically, just as precise and
accurate, but their connection to the real world is far more tenuous and unreliable – as
has been demonstrated several times in recent years when those mathematical results
have resulted in financial crises, and occasionally disasters.

So what of the paradox of voting? The paradox arises when you start by assuming that
people vote to choose, say, a president. Yes, we all say that is what we do. But that’s
just because we have drunk Election Kool-Aid. We don’t actually behave in accordance
with that statement. If we did, then as rational beings we would indeed stay at home on
election day.

Time to throw out the Kool-Aid and buy a gallon jug of far more beneficial Election Milk:
(Presidential) elections are about a society choosing a president. Where that purpose
impacts the individual voter is not who we vote for, but in providing social pressure to
be an active member of that society.

That this is what is actually going on is illustrated by the fact that U.S. society created,
and millions of people wear, "I have voted" badges on election day. The focus, and the
personal reward, is not "Who I voted for" but "I participated in the process." [For an
interesting perspective on this, see the recent article in the SmithsonianMagazine, "WhyWomen Bring Their “I Voted” Stickers to Susan B. Anthony’s Grave."]

To be sure, you can develop mathematical models of group activities, like elections, and
they will tend to lead to fewer problems (and "paradoxes") than a single-individual model
will, but they too will have limitations. All mathematical models do. Mathematics is not
reality; it is just a model of reality (or rather, it is a whole, and constantly growing,
collection of models).

When we develop and/or apply a mathematical model, we need to be clear what
questions it is designed to help us answer. If we try to apply it to a different question, we
may get lucky and get something useful, but we may also end up with nonsense,
perhaps in the form of a "paradox."

With both measurement and the election, as is so often the case, one benefit we get
from trying to apply mathematics to our world and to our lives is we gain insight into
what is really going on.

Attempting to use the real numbers to model the acts of measuring physical objects
leads us to recognize the dependency on the physical activity of measurement.

Likewise, grappling with Downs Paradox leads us to acknowledge what elections are
really about – and to recognize that choosing a leader is a societal activity. In a
democracy, who each one of us votes for is inconsequential; that we vote is crucial.
That’s why I did not just spend a couple of hours yesterday making my choices and
filling in my ballot and leaving it at that. I also went out earlier today – in light rain as it
happens (and without a sore throat) – and put my ballot in the mailbox. Yesterday I
acted as an individual, motivated by my felt societal obligation to participate in the
election process. Today I acted as a member of society.

As a professional set theorist, I am familiar with the relationship between, and the
distinction between, a set and its members. When we view a set in terms of its
individual members, we say we are treating it extensionally. When we consider a set in
terms of its properties as a single entity, we say we are treating in intensionally. In an
election, we are acting intensionally (and intentionally) – at the set level, not as an
element of a set.

* A shorter version of this article was published simultaneously in The Huffington Post.

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The Mathematical Association of America is the world's largest community of mathematicians, students, and enthusiasts. We accelerate the understanding of our world through mathematics, because mathematics drives society and shapes our lives. Visit us at maa.org.